Liquid Madelung energy accounts for the huge potential shift in electrochemical systems

Achievement of carbon neutrality requires the development of electrochemical technologies suitable for practical energy storage and conversion. In any electrochemical system, electrode potential is the central variable that regulates the driving force of redox reactions. However, quantitative understanding of the electrolyte dependence has been limited to the classic Debye-Hückel theory that approximates the Coulombic interactions in the electrolyte under the dilute limit conditions. Therefore, accurate expression of electrode potential for practical electrochemical systems has been a holy grail of electrochemistry research for over a century. Here we show that the ‘liquid Madelung potential’ based on the conventional explicit treatment of solid-state Coulombic interactions enables quantitatively accurate expression of the electrode potential, with the Madelung shift obtained from molecular dynamics reproducing a hitherto-unexplained huge experimental shift for the lithium metal electrode. Thus, a long-awaited method for the description of the electrode potential in any electrochemical system is now available.


S1. Liquid Madelung potential calculation
In this study, ELM is obtained numerically by calculating the Coulombic interaction energies between the target Li + ion and other solvents/ions during the MD simulations under periodic boundary conditions and subsequently averaging these values over time for all Li + ions as follows: where E tot is the total Coulombic energy of the system,   extracted is the Coulombic energy of the system with the i-th Li + extracted, E Li+ is the Coulombic energy of the system with one Li + , and NLi+ is the number of Li + .Note that the Coulombic energies of the charge excess systems (E extracted and E Li+ ) under periodic boundary conditions include the numerical errors due to the background charges (BC) to neutralize the system, depending on the simulation cell size.To mitigate this unavoidable effect, the simulation cell size was kept almost the same for each electrolyte.The errors are indeed negligible, because the calculated potential shifts with BC showed almost identical values with those obtained by directly summing the Coulombic interactions (without BC) among Li + and other solvents/ions up to the electrostatically shielded distance (i.e., 30 Å) (Fig. S11).

S2. Theoretical basis
Eq. [1] in the main text is converted as follows: where µLi+ is the chemical potential (or partial molar Gibbs energy) of Li + .Here, the chemical potential is rewritten using the partial molar enthalpy and partial molar entropy, and the enthalpy term can be described as the sum of ELM for Li + and the other terms: Therefore, the difference in the chemical potential is given by If ΔELM >> ΔEother -TΔS, or ΔELM is the predominant contribution to the enthalpy term and the entropy term is relatively small, ΔELi/Li+ can be expressed as follows: In Figure 3a-c

Fig. S3 :
Fig. S3: Experimental potential shifts (ΔEZn/Zn2+ (Exp.)) and the calculated ones from liquid Madelung potential (ΔELM/F) in Zn(TFSI)2/PC electrolytes as a function of salt concentration mZn2+ with reference to 0.25 mol kg -1 .The black dashed line represents the potential shift based on the ideal Nernst equation, where γzn2+ is assumed to be unity.

Fig. S7 :
Fig. S7: Electron density isosurface mapped with molecular electrostatic potential of the PC and SL solvents (carbon, oxygen, hydrogen and sulphur shown in grey, red, white, and yellow, respectively).

Fig. S9 :
Fig. S9: Experimental data for ΔELi/Li+ and the calculated potential shifts of liquid Madelung potential (ΔELM/F) defined as the shift from the data for 1.5 mol dm -3LiFSI/THF electrolyte without toluene and plotted as a function of the ratio of toluene to the total solvent amount (%).